I had loads of work to take care of, but I recently had time to take a closer at this problem.
It is my current understanding that the authors of the paper use the quadrupole moment of mass distribution because it is a mathematical object easily converted to homogeneous coordinates, whereas I have not yet found an homogeneous coordinates equivalent for the inertia tensor.
For the mass quadrupole
M, using the definition:
<br />
M = \int_V{\rho(\vec{u})\vec{u}\vec{u}^t dx dy dz}<br />
(where \rho(\vec{u}) is the density at point \vec{u}) with a vector \vec{u} in euclidian space:
<br />
\vec{u} = \left(\begin{array}{c} x \\ y \\ z \\ \end{array} \right)<br />
you have the matrix form:
<br />
M = \int_V{\rho(\vec{u})\left(<br />
\begin{array}{ccc}<br />
x^2 & xy & xz \\<br />
xy & y^2 & yz \\<br />
xz & yz & z^2 \\<br />
\end{array}<br />
\right)<br />
dx dy dz}.<br />
And if you consider \vec{u} to be a vector in projective space (homogeneous coordinates):
<br />
\vec{u} = \left(\begin{array}{c} x \\ y \\ z \\ 1 \\ \end{array} \right)<br />
you have the matrix form:
<br />
M = \int_V{\rho(\vec{u})\left(<br />
\begin{array}{cccc}<br />
x^2 & xy & xz & x\\<br />
xy & y^2 & yz & y\\<br />
xz & yz & z^2 & z\\<br />
x & y & z & 1\\<br />
\end{array}<br />
\right)<br />
dx dy dz}.<br />
Since the definition is given in terms of vector product, it can be applied in homogeneous coordinates just as in cartesian coordinates.
Now consider the transformation matrix
W in homogeneous coordinates that changes the reference frame of
M (through rotation and translation):
<br />
W = \left(<br />
\begin{array}{cccc}<br />
& & & \Delta_x\\<br />
& R & & \Delta_y\\<br />
& & & \Delta_z\\<br />
0 & 0 & 0 & 1\\<br />
\end{array}<br />
\right)<br />
where
R is a 3x3 rotation matrix and \Delta_i is the translation along dimension
i. We have that WMW^t correctly expresses
M in homogeneous coordinates in the new reference frame (the proof is left as an exercice to the reader
). One could adapt the parallel-axis theorem to the mass quadrupole and see that this matrix multiplication in homogeneous coordinates generalizes it (by allowing rotation as well as translation).
Now, on the other hand, the moment of inertia tensor
I is not really defined in terms of vectors. The most general definition I have found is:
<br />
I = \int_V{\rho(\vec{u})\left(\|\vec{u}\|^2E_3 -\vec{u}\vec{u}^t\right) dx dy dz}<br />
where E_n is the
n x
n identity matrix. Considering \vec{u} as a vector in euclidian space, you get the usual matrix form:
<br />
I = \int_V{\rho(\vec{u})\left(<br />
\begin{array}{ccc}<br />
y^2+z^2 & -xy & -xz \\<br />
-xy & x^2+z^2 & -yz \\<br />
-xz & -yz & x^2+y^2 \\<br />
\end{array}<br />
\right)<br />
dx dy dz}<br />
which is fine, but if you want to express it in homogeneous coordinates (which requires replacing the 3x3 identity matrix by the 4x4 identity matrix in the definition), you get the following matrix form:
<br />
I = \int_V{\rho(\vec{u})\left(<br />
\begin{array}{cccc}<br />
y^2+z^2+1 & -xy & -xz & -x\\<br />
-xy & x^2+z^2+1 & -yz & -y\\<br />
-xz & -yz & x^2+y^2+1 & -z\\<br />
-x & -y & -z & x^2+y^2+z^2\\<br />
\end{array}<br />
\right)<br />
dx dy dz}<br />
or, if you consider that \|\vec{u}\|^2 = x^2 + y^2 + z^2 instead of \|\vec{u}\|^2 = x^2 + y^2 + z^2 + 1,
<br />
I = \int_V{\rho(\vec{u})\left(<br />
\begin{array}{cccc}<br />
y^2+z^2 & -xy & -xz & -x\\<br />
-xy & x^2+z^2 & -yz & -y\\<br />
-xz & -yz & x^2+y^2 & -z\\<br />
-x & -y & -z & x^2+y^2+z^2-1\\<br />
\end{array}<br />
\right)<br />
dx dy dz}.<br />
However, for both forms, WIW^t fails to correctly express the moment of inertia tensor in the new reference frame. In particular, applying a translation using
W does not give the same result as applying a translation using the parallel-axis theorem. At the time being, I don't see what form the moment of inertia tensor should take in homogeneous coordinates to remain valid under rigid transformations (rotation & translation).
In my opinion, the previous (partial) demonstration justifies the use of the mass quadrupole instead of the moment of inertia tensor for use in multibody dynamics in homogeneous coordinates (as changing reference frame is a common task in that context).
Therefore my initial question remains: If anyone knows of a reference book on the subject of multibody dynamics in homogeneous coordinates, please let me know! And if such a book uses the moment of inertia tensor rather than the mass quadrupole, I would be quite interested in seeing the form of the inertia tensor in homogeneous coordinates!
